In the **previous post**, we have learned the graphical representation of domain and range. The domain of the function is the shadow or projection of the graph of to the x-axis (see the red segment in the figure below). The range of is the projection of the graph of to the y-axis (see the green segment in the figure below). In this post, we are going to learn how to analyze equations of **functions** and determine their domain and range without graphing.

If a graph of a function is projected to the x-axis, the projection is the set of x-coordinates of the graph. A single point on the projection means a point on the graph exists. The existence of a point implies that exists. This means that the function is defined at . In effect, the domain of a function is the set of x-coordinates that makes the function defined. In what follows, we learn some examples to illustrate this concept.

**Example 1:** Find the domain and range of .

*Explanation*

As we can see from the equation, we can substitute any real number to and any of these values makes the function defined. Therefore, the domain of the function is the set of **real numbers**. The range of the function has also no restriction. We can think of any real number , and we can always find the corresponding by subtracting 1 from it. Therefore, the range of is the set of real numbers.

**Example 2:** Find the domain and range of .

*Explanation*

Just like in Example 1, we can substitute any real number to and any of them makes defined (Can you see why?). So, the domain of is the set of real numbers.

Now let’s think about the range. The range are the values of . Notice that if we square a positive real number, the result is a positive real number. If we square a negative real number, then the result is also a positive real number. If we square 0, the result is 0. Therefore, the only possible results that we can get if we square a real number are 0 and positive real numbers. Therefore, the range of is the set of real numbers greater than, or equal to 0. In interval notation, the range of is .

**Example 3:** Find the domain and range of

Recall that means the absolute value of . As we can see, we can substitute any real number to , and is always defined. So, the domain of is the set of real numbers.

As for the range, if is a negative real number, then is a positive real number. If is , is also . If is a positive real number, then is a positive real number. Since all values of is either 0 or positive real numbers, the range of , is the set of real numbers greater than or equal to 0. In interval notation, the range of is .

**Example 4:** Find the domain and range of

*Explanation*

We know that we cannot find a real number that is a square root of negative numbers. This means that we cannot substitute a negative number for . In effect, we limit the values of to 0 and positive real numbers. Therefore, the domain of is the set of real numbers greater than or equal to 0. In interval notation, the domain of is .

Since we are only allowed to substitute and positive real numbers to , then, it follows that will also result to 0 or a positive real number. Therefore, the range of is the set of real numbers greater than or equal to 0. In interval notation, the range of is .

**Example 5:** Find the domain of

*Explanation*

We know that we cannot have a zero denominator because it will make the fraction undefined, so cannot equal to 0. But aside from 0, we can substitute any value to , so, the domain of is the set of real numbers except 0. In interval notation, the domain of is .

For the range, examine the equation . We can get that are small by substituting large values to . We can also get that are large by substituting small values to . We can also choose an that will make negative. But we cannot get a value of , because the only way to do this is for the numerator of the expression to be 0. But our expression is , so cannot be 0. Therefore, the range of is the set of real numbers except 0. In interval notation, the range of is is .

In the **next post**, we will have more examples on how to determine the domain and range of a function.